K. Borsuk in [1]} introduced concepts of the capacity and the depth of a compactum together with some relevant problems. Indeed, the capacity \(C(A)\)  of a compactum \(A\)  is the cardinality of the class of the shapes of all compacta \(X\)  such that \(Sh(X)\leqslant Sh(A)\) , and the depth \(D(A)\)  of a compactum \(A\)  is the least upper bound of the lengths of all chains \(\mathcal {S}h(X_{k}) <\cdots < \mathcal {S}h(X_{2}) < \mathcal {S}h(X_{1}) \leqslant \mathcal {S}h(A)\) , where \(\mathcal {S}h(X)<\mathcal {S}h(Y)\)  denotes that \(\mathcal {S}h(X)\leqslant \mathcal {S}h(Y)\)  holds but \(\mathcal {S}h(Y)\leqslant \mathcal {S}h(X)\)  fails. If this upper bound is infinite, we write \(D(A)=\mathcal {N}_0\)  (for more details, see [2]}). It is clear that \(D(A) \le C(A)\)  for each compactum \(A\) .
